The aim of this paper is to show that every representative function of a maximally monotone operator is the Fitzpatrick transform of a bifunction corresponding to the operator. In fact, for each representative function ϕ of the operator, there is a family of equivalent saddle functions (i.e., bifunctions which are concave in the first and convex in the second argument) each of which has ϕ as Fitzpatrick transform. In the special case where ϕ is the Fitzpatrick function of the operator, the family of equivalent saddle functions is explicitly constructed. In thiswaywe exhibit the relation between the recent theory of representative functions, and the much older theory of saddle functions initiated by Rockafellar.

Bianchi, M., Hadjisavvas, N., Pini, R., Representative functions of maximally monotoneoperators and bifunctions, <<MATHEMATICAL PROGRAMMING>>, 2018; 2018 (168): 433-448. [doi:DOI 10.1007/s10107-016-1020-8] [http://hdl.handle.net/10807/78681]

Representative functions of maximally monotone operators and bifunctions

Bianchi
Primo
;
2016

Abstract

The aim of this paper is to show that every representative function of a maximally monotone operator is the Fitzpatrick transform of a bifunction corresponding to the operator. In fact, for each representative function ϕ of the operator, there is a family of equivalent saddle functions (i.e., bifunctions which are concave in the first and convex in the second argument) each of which has ϕ as Fitzpatrick transform. In the special case where ϕ is the Fitzpatrick function of the operator, the family of equivalent saddle functions is explicitly constructed. In thiswaywe exhibit the relation between the recent theory of representative functions, and the much older theory of saddle functions initiated by Rockafellar.
Inglese
Bianchi, M., Hadjisavvas, N., Pini, R., Representative functions of maximally monotoneoperators and bifunctions, <<MATHEMATICAL PROGRAMMING>>, 2018; 2018 (168): 433-448. [doi:DOI 10.1007/s10107-016-1020-8] [http://hdl.handle.net/10807/78681]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/78681
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